In this paper, we consider the following problem \(\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^s u =\lambda \vert u\vert ^{q-2} u+\left( \displaystyle \int _{\Omega } \frac{\vert u(y)\vert ^{2_{\mu ,s}^{*}}}{\vert x-y\vert ^{\mu }} d y\right) \vert u\vert ^{2_{\mu ,s}^{*}-2} u, & \text{ in } \Omega , \\ u =0, & \text{ on } {\mathbb {R}}^{N}\backslash \Omega , \end{array}\right. \end{aligned}\) where \(\Omega \) is an open bounded set with continuous boundary in \({\mathbb {R}}^{N}(N \ge 4s)\) , \(0<\mu <N\) , \(2_{\mu ,s}^{*}=\frac{2 N-\mu }{N- 2\,s}\) and \(q \in \left[ 2,2_s^{*}\right) \) where \(2_s^{*}=\frac{2 N}{N- 2s}\) . Using the Nehari manifold and Ljusternik-Schnirelmann category, we relate the number of positive solutions of the above problem to the topology of \(\Omega \) .
